THESIS 222 222 22 2222222 \2 l/ M..- This is to—eertfi‘flhat the k i ii Fhes‘i‘s’ Efifftled “ g: l ‘l vu"~.—vm‘— l..*—. a —.o—v .... .7..-J -—. -..~ ~--M—”-* —--—”- . o. “.1--- MicrowaJe Q§cill§tor Simulation -oH---- --——.—- .- ---.—n—.-.a.'. 4--..-- presented by Michael J. Schnars has been accepted towards fulfillment of the requirements for M.S. degree in Elec. Engr. fivc’ggm» Major professor Date Feb. 25, 1983 0-7639 MS U is an Afirmau've Action/Equal Opportunity Institution PLACE ll RETURN BOX to remove this checkout {tom your record. TO AVOID FINES Mum on or before due out DATE DUE DATE DUE DATE DUE MSU Is An Nflrmdlvo ActionIEqml Oppommlty Institution . cmmw‘ MICROWAVE OSCILLATOR SIMULATION BY Michael John Schnars A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering and System Science 1983 ABSTRACT MICROWAVE OSCILLATOR SIMULATION BY Michael John Schnars A new technique of analysis of the waveguide-coaxial IMPATT diode oscillator is presented. The circuit is modeled as sections of transmission line with shunt admit- tance elements representing the circuit tuning screws and coupling apertures. The circuit models are used as elements of ABCD matri- ces and the circuit analysis is performed by computer. The computer program provides data lists of the circuit variable values, plots, or a complete statistical analysis of the circuit data. The program was written to allow easy modifi- cation of the models used, the circuit structure. or variation of a single circuit parameter. The program also allows circuit losses to be varied, and mismatches at the load or stabilizing resistor can be varied to determine their effect on circuit performance. II. III. IV. VI. VII. VIII. IX. X. XI.. TABLE OF CONTENTS Page List of Tables .............................. ii List of Figures ............................. iii Introduction ................................. v Benefits and Limitations of Modeling ......... 1 Computer Program Format ...................... 3 ABCD Matrix Models ........................... 6 Circuit Analysis ............................. 10 Data Analysis ................................ 15 Validity Of MOdeling OOOOOOOOOOOOOOOOOOOOO0.0. 19 Future Enhancement ........................... 20 summary 0.00..0.000......OOOOOOOOOOOOOOOOOO0.0 22 References ODOOOOOOOOOOOQOO0.0000000000DOOOOOO 23 Appendices OI.O0.00.00...OOOOOOOOOOO0.000000000 24 Appendix 1 Circuit and Control Variables .. 24 Appendix 2 Program Arrays ................. 29 Appendix 3 ABCD Matrix Models ............. 32 Appendix 4 Quartz Rod Analysis ............ 40 Appendix 5 Attenuation, Phase Factor and Impedence Equations ...... 42 Appendix 6 Device Circuit Angle ........... 45 Appendix 7 Variable Effects ............... 49 Appendix 8 Program Description ............ 53 Appendix 9 Data Sample .................... 58 Appendix 10 PIOtS 0..OOOOOOOOOOOOOOOOOOOOOOO 65 Appendix 11 Program Listing ................ 81 Table Table Table Table Table Table Table Table Table Table Table Table WIBWNH 11 12 LIST OF TABLES Planes Used for Circuit Division ............ Program Control Variables ................... ABCD Matrix Models .......................... ABCD Matrix Equations ....................... Variables That Parameter has Dominant Effect 0n (Mean Value) Variable Least That Parameter Effects By At 0.1% (Mean value)OOOOOOOOOOOOOOOOOO Variable Least That Parameter Effects By At 0.1% (Standard Deviation) Variable That Parameter Has Dominant Effect 0n (Standard Deviation) Paramater values 0 O O O O O O O O O O O O O O O O O O O O O O O O O O O variable values O O O O O O O O O O O O O O O O O O O O O O O O O O O O O StatiStical value O O O O O O O O O O O O O O O O O O O O O O O O O O O Best Statistical Values ..................... ii 49 50 51 52 58 59 62 64 Figure Figure' Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure u: ‘0 GD '4 0‘ Ln :5 ll 12 13 14 15 16 17 18 19 20 21 22 23 24 LIST OF FIGURES Side View Of OSCillator OOOOOOOOOOOOOOOOOOOOO TOP View Of OSCillator OOOOO‘OOOOOOOOOOOOOOOOO Oscillator Equivalent Circuit ............... ABCD Voltages and Currents .................. Tuning screw M0d91 OOOOOOOOOOOOOOOOOOOOOOOOOO Coupling Pad "Odel OOOOOOOOOOOOOOOOOOOOOOOOOO Step CapaCj-tor "Odel OOOOOOOOOOOOOOOOOOOOOOOO Coupling Pad Inward Reduction Model ......... Coupling Pad Outward Reduction Model ........ Quartz Rod "Odel OOOOOOOOOOOOOOOOOOOOOOOOOOOO Page 24 24 25 37 37 38 38 39 39 40 DeVice Angle OOOOOOOOOOOOOOOOOOOOOOOOOOOOO 46' 47 Reflection Reflection Reflection Reflection Real Power Real Power Y Imag. as Y Imag. as Coefficient as LTR Varies ........ coeffiCient as r1 varies ooooooooo Coefficient as d1 Varies ......... Coefficient as Y8 Varies ......... as r1 varies O O O O O O O O O O O O O O O O O O O O O as a varies O O O O O O O O O O O O O O O O O O O O O O Lx varies O O O O O O O O O O O O O O O O O O O O O O O O QROd varies OOOOOOOOOOOOOOOOOOOOO YReal as Lx varies OOOOOOOOOOOOOOOOOOOOOOOOO YReal aSQROd varies OOOOOOOOOOOOOOOOOOOOOO Eff. asavaries OOOOOOOOOOOOOOOOOOOOOOOOOOOO Eff. as d1 varies OOOOOOOOOOOOOOOOOOOOOOOOOOO QL as Lx varies ooooooooooooo0000000000000... iii 65 66 67 68 69 70 71 72 73 74 75 76 77 Page Figure 25 CL as Q ROd varies 00000000..0000000000000... 78 Figure 26 SS as Iris varies OOOOOOOOOOOOOOOOOOOOOOOOOOO 79 Figure 27 SS as Q Rod Varies .........-.,.............. 80 iv INTRODUCTION The use of an IMPATT diode as a microwave power source requires considerable design ingenuity to ensure that the oscillator is stable and that spurious oscillations cannot occur as a result of reflections from either the D.C. bias filter or output mismatches. To aid in the design analysis, a new computer simulation of the oscillator has been deve- loped. Rather than modeling the circuit as lumped elements as was previously done, (Ref. 10, 11, 12, 13), the circuit is modeled using ABCD parameters, with each element of the circuit modeled as one ABCD matrix. This allows great flexibility in the choice of models for each element of the circuit, from sections of transmission line, to data values obtained from experimental measurement of the circuit ele- ment. The circuit used for the oscillator is a general single-tuned, waveguide—coaxial circuit sometimes called a Kurokawa-type resonator (Ref. 14). The IMPATT diode is biased into reverse avalanche breakdown, and because of the transit time of the charge carriers through the diode, it acts as a negative resistance, which, in an appropriate cir- cuit, will cause oscillation to occur. The configuration of the circuit is easily changeable to allow comparison of dif- ferent designs. The circuit parameters (radius, etc.) can be changed over a wide range of values, or lossy materials can be introduced into the circuit to see the effect on the overall circuit performance. The simulation is performed at V vi 20 frequencies over the range of 17.7-19.7 GHZ. The program calculates the circuit efficiency, the cavity Q's, relative load power, input admittance, reflection coefficient, and the frequency derivative of the susceptive part of the input admittance (susceptance slope). The program can provide plots of these variables, or lists of their data points over the frequency range, or analyze the data points and provide mean value, standard deviation, and several other measures of the central tendency and variability of the distributions of the circuit variables. Work of this type is urgently needed to aid in efforts to develop power combiners using 4 or 6 IMPATT diodes. An accurate model of each individual oscillator is required to allow predictions of the response of each oscillator as its load admittance is changed as they are combined, and from variations in the IMPATT diodes from different production runs. Benefits and Limitations of Modeling The major benefits of computer modeling of the diode and resonant structure come from the speed of execution of the program. Analyzing the circuit by hand calculations can take several hours at each frequency. The program provides a convenient method to analyze a new design, or to compare several different designs by simply changing the appropriate ABCD matrix. The program can also be used to analyze and improve existing design. If, for example, lower noise is required, the information on the movement of the device line (Appendix 6) can be used to improve the device angle for lower noise. In this complex circuit, varying a single parameter may affect several different aspects of the cir- cuit. For example, increasing the coupling aperture (d1) increases the input admittance and magnitude of the reflec- tion coefficient but decreases the cavity 0, 00. LTl (the tuning screw waveguide length), however, can be used to increase 00 so as to compensate for the actions of d1. These type of trade-offs are easily viewed, using the results of the computer simulation. The graphs generated by the program can also be used to find when 'loops' will form in the reflection coefficient locus (indicating possible instabilities in the operating point) while tuning the device. The effect of losses on circuit performance can be predicted and compared to experimental results. This could be used to determine if plating the inside of the resonant cavity to increase the 'Q' is worth the extra expense. Also, since the program can be used to determine what the optimum value and variation of the device impedance should be, it would be possible to change the package of the IMPATT diode to match the impedance the circuit can present when the stabilizing loss is small. The model of the oscillator has several inherent assumptions and limitations. In order for the analysis to be meaningful, the assumption that the evanescent fields of obstacles do not interact must be made, and over moding in the cavity due to the quartz rod is not included. The circuit is assumed to be reciprocal, and all restrictions on variable range indicated in the section on the ABCD models must be met. Computer Program Format The program is written in Fortran 4 and is 2000 lines long. Fortran was chosen because this provides compati- bility with other computer installations, and allowed usage of the CalCom plot routines. Also, the large number of external and intrinsic functions, and the array manipulation commands of Fortran make this language a natural choice. This program was written to help in the design and ana- lysis process of microwave oscillators. Shown in Appendix 1, Figures 1 and 2 are side and top views of the specific oscillator circuit chosen for simulation and the names of the circuit parameters. Figure 3 shows an equivalent cir— cuit of the oscillator and indicates the circuit sections the models represent. The planes shown in Figure 3 and listed in Table 1 partition the equivalent circuit into cells, each of which is represented by one ABCD matrix. The circuit is analyzed by starting at the load and finding the admittance looking toward the load at each successive plane until the admittance the IMPATT diode would see at plane 1 is found. Then, by assuming the diode acts as a l-volt source at plane 1, the voltage and current is found at each plane out to the load. The analysis is then repeated at each frequency desired. Except for the model used for the quartz rod, the program can be run at any frequency desired, using any increment size between frequencies (up to 20 frequencies per run total). The currently used model for the quartz rod, however, restricts the program to 17.7-19.7 GHZ and 0.1 GHZ step sizes. This is because the model for the quartz rod was determined from experimental data, and the data was taken at these points. The program is set up so that by changing the value of a flag variable from '0' to '1' at the start of the program, any desired combination and number of parameters can be used each run, and any combination of the possible types of outputs can be selected. Appendix 1, Table 2 lists the variables used in controlling the program actions, and Appendix 2 lists the arrays used in the program and typical values of the circuit parameters. To use the program, the parameters to be varied are first selected (for example, the length of the 1/4 wave transformer, and the cavity height). The program will analyze the effect of the variation of each parameter separately. Next, the range of the variables are chosen. The variables can have any value, with up to 10 increments of any size in each run. The type of output desired is selected next. For initial runs, outputting only the data indicates the variation of the distribution of the circuit variables values, gives good indications of trends, is very cheap, and avoids the large amount of data that would be generated if the actual value of the load power, etc., were output at each frequency for each value of each variable. After initial runs, the points of maximum effectiveness of each variable can be determined, and the range narrowed to the region of interest. For final runs, data lists and graphs with multiple curves can be output for complete ana- lysis. The program description is contained in Appendix 8, and Appendix 11 has the program listing which indicates in the symbolic reference map what the program variables names mean. ABCD Matrix Models The ABCD or 'cascade' parameters were chosen because in the analysis of the circuit, the input voltages and currents are transferred through each circuit model to find the out- put voltages and currents, and the input admittance to each model is also easily found with these parameters. In addi- tion, the ABCD parameters provide ease of the modification of the model of each element, and (under the reciprocity assumption) the inverse of the 2 x 2 matrix is easily calcu- lated — providing great computational advantage in this case. Also, the ABCD parameters allow easy incorporation of the frequency variation of the models used for the admit- tance elements. The ABCD models used for the computer simulation are shown in Appendix 3, Table 3. These were deve10ped from Reference 17. Table 4 shows the ABCD elements used for each section of the model. Figure 4 shows the voltages and currents defined for the ABCD model. For the circuit model, a total of 18 ABCD matrices were required. Of these, 7 are admittance models and 11 are sec- tions of transmission line. For the development of the model, the load was set to the value of the characteristic admittance of the output waveguide - i.e., we consider no mis-match at the load for simplicity. One may include these reflections if desired. The model used for the metallic tuning screws is a T-pad of impedences Ref. (1) (Fig. 5). The corresponding 2 Z Z Z Z Z Z ABCD elements are A = 1 + 1 B — 1 2 + g 3 + 1 3 Z3 - 2 C = -—l— D = 1 + 2 . In our case, ~21 = - jxa and z3 z3 22 = 21 and 23 = — ij. Then the elements B and D become .113 B = Z + 2 21 and D = 1 + 21/23. The reactances Xa and Kb 3 are defined as xa = 20' (1.55 x 10'3t) and Xb = zo'(3.9525/t) where t is the depth of the tuning screw, and can vary from 0 to 0.12 inch. The values of the constants were determined from the graphs of experimental data in Ref.(1). With this model, the tuning screws are used to tune for maximum power. Two metallic capacitive tuning screws are used so the one closest to the iris (at a distance of Xg/B) will act as a capacitor in parallel with the iris. The second screw (at a distance of .Ag/4) will act as an inductor in parallel with the iris. Since the iris is modeled as an inductance (Ref. 2), the first tuning screw effectively increases the iris diameter, and the second effectively decreases the iris diameter. The elements of the iris ABCD matrix are: A = D = l, B = 0 and C = - jBi where 2-327A9——a§—— (d3 is the iris diameter and a and b are the waveguide dimensions). This model is valid in the wave- length range 2a ) A) a and for the diameter of the iris <;n A = D = 1, B = O, C = Yq where Yq = 1 + where pin Din is the reflection coefficient measured experimentally. We assume only T810 (H10 mode and guide cutoff is 14.1 GHZ. The coupling between the air dielectic coaxial line and the resonant cavity (through the oblong aperture) is modeled by a fi-pad (Ref. 3). The elements of the pad (Fig. 6, Appendix 3) are defined as Ba = Y0 P/(2310r221n(r2/r1)) and d d 24E! 6) and M = d13— “ E 24 (Ne) - 3(a)) and E "' (1 " (dz/d1)2 )1”- Bb = (Yo Aorzzlnuz/rln/M where p F( e) and E( e) are complete elliptic integrals of the first and second kind, respectively. From an analysis of M and P (Ref. 4), these functions can be represented over the range of interest by a straight-line approximation, which is used by our model. This model is valid for d1 << 4, and for the wavelength range for which only the principal mode can be propagated in the coaxial guide and only the H10 mode in the rectangular guide. It is also assumed the curvature of the coax containing the aperture has negligible effect, and the coax outer wall is assumed to have zero thickness. The step discontinuity (Fig. 7, Appendix 3) in the coaxial radius is modeled as a capacitance (Ref. 5). The defining equation is Y step = 93:5 x 1012 - where f is the frequency in Hz. This is valid over the entire range of radii of interest. The remainder of the sections of the oscillator are modeled as sections of transmission line. The ABCD elements are A cosh (Y'L), B = sinh (Y'L)/ YOL' C = YOL sinh (yL) and D cosh (Y'L) where y' = cx+ jB and L and YOL are the length and characteristic admittance of the line in question. The values of a. and B are different in the air dielectric coax, the teflon dielectric coax, and the waveguide. The specific equations used in each case are recorded in Appendix 5. The method used in determining the admittance in the waveguide is covered in Appendix 5. Circuit Analysis The analysis of the circuit is easily effected using the ABCD parameters as previously described. Starting at the load, the input admittance to each section is found by using Yin = g :Eg;%¥i; . This is then combined with the load admittance of the next section (if any). This is the new load admittance of the next section, and the next input admittance is then found as described above. The circuit is reduced in this fashion from plane 18 to 9, and from 19 to 7 so the circuit is reduced into the coupling pad (Fig. 8, Appendix 3). The input admittance of the coupling pad is . 6-19 _ - . 9-18 _ - then found as Yin = ::?n6-I9 _-%: )iyy. 9_ § ) - jBa. in a in Then the reduction down to the diode is performed by finding the input admittance to each section using the ABCD parameters. As the circuit is reduced, the input admittance at each plane is stored in an array, and the reflection coefficient at each plane is calculated using a Yin " YAL pin Yin + YoL and the value is saved. After the above reduction is performed at each desired frequency, an array of 20 values (one for each frequency) is obtained for the input admittance and reflection coefficient as seen by the diode and can be an output (the values of the admittance and reflection coefficient at each plane can also be an output). At the diode plane, once the input admittance is 10 11 known, we use a l-volt generator to represent the diode. The current is then calculated using 11 = YinVl (V1 = 1 volt) and the input power is Pin = L2 RE (V11*1). The next step in the analysis is to find the voltage and current at each plane using [‘12:] = [g ';_l E111] (as defined in Fig. 4, Appendix 3). The voltage and currents are found from plane 1 to 5 (the coupling pad). At this point a new ABCD matrix is defined. Since we are not interested in the voltages and currents in the stabilizing load, we define a matrix to transfer the voltages and currents from plane 5 to plane 9 (Fig. 9, Appendix 3). A = 1 + - 38? = l . Ystab ' 33a ' Ystab ‘ 33a C = - jBa ‘ 33b + ( I‘lBa)( - 7gb) and D = 1+"“4:‘l§aT‘-‘o Ystab ‘ 33a) (Ystab ' 33a) The remaining voltages and currents out to the load can now be found using the remaining ABCD matrices. At the load, the power is found as PL 8 L2 RE (V18I*13) and the circuit efficiency is found as nckt = §§:-. The voltage and current at each plane is saved, and can be an output. Since all circuit variables are now known, we can calculate a 'secondary' set of variables. First, by defining the values of admittance the diode needs to work into (the device window) as 0.93 g_| pI | 5_ 0.985 and -35°‘g_ 01 §_.- 45° (based on experimental work), we can find the angle of intersection between the device line (assumed radial) and the locus of the input admittance. This angle is the device angle, and the points of the input 12 admittance contained in the device window is the ”effective bandwidth“ of the circuit. The program also determines the frequency which is nearest the device window, if none are in it. (A complete analysis of the device angle is contained in Appendix 6.) Since the circuit input admittance is known, we can find the input susceptance slope using Yin = Gin + j Bin so that the susceptance slope = BB in/Bw . 1 Also, the cavity Q's are QL = O and also ZRE (Yin) = ———QL—-——— and = ( —L_ _ l )-l 09" n ckt 0° 01, 0.x ° At this point, the circuit has been completely charac- terized. All values of all variables at each point in the circuit are known. However, since 33 circuit parameters can be varied in 10 increments each run, and analyzed at 20 fre- quencies for each value of each increment of each variable, it is easy to generate massive amounts of data. To facili- tate the analysis of the data, the program provides one final aid to analysis. For each of the variables of major interest (efficiency, load power, input admittance, magni- tude of reflection coefficient, angle of reflection coef- ficient, susceptance slope, QL. 00, and Qex), the program analysis distribution variability. The program considers each parameter being varied separately, and uses the first value of the parameter as a reference point. Then, for each increment of the parameter, the program calculates the mean values, standard deviation, number of points larger than the reference, the average value of increase of the points .255:- 13 larger than the reference, the number of points less than the reference, the average value standard deviation, and mean value of decrease of the points less than the reference. The program also determines which increment of the parameter has the largest value, and which has the smallest value over the frequency range, and the frequency at which they occur. (The information on the smallest values can be used, for example, to decrease the complex part of the load power if the load is mismatched.) For each of these statistical values, the program can either print them directly or compare them and print which increment has the best value, the percent increase over the reference, the value of the circuit parameter at that increment and the best value itself. If, for example, the depth of a tuning screw is being varied, then the program can analyze the cir- cuit and tell what value of the tuning screw (if any) gave the largest increase in the mean value of the efficiency (for example), the percent increase, and what the largest value of the mean was, and which increment of the tuning screw gave it (Table 12, Appendix 9). Since only four num- bers are output (for each statistical variable -- mean, standard deviation, etc., and each circuit variable -- effi- ciency, load power, etc.), two pages of data provides all information of interest from ten increments of a variable. Thus, every parameter in the circuit can be varied and only 70 pages of data are generated, and it is not necessary to know every value of the circuit variables at every frequency 14 to analyze the circuit. This is of great aid when trying to increase the mean value of the load power, for example, since four numbers indicate which value of the variable gave the best results. Also, the design trade-offs are imme- diately obvious since the statistical information indicates whether the efficiency (for example) was affected or not. Data Analysis Since the program was designed to allow optimization of the circuit for a particular application (i.e., maximum bandwidth, or efficiency, etc.) or to compare several dif- ferent circuit designs, no ‘best' values for the circuit parameters can be given. However, by analyzing the data generated by the program over typical ranges of the parame- ters, it is possible to determine the effect each of the parameters has on the circuit. In order to see general trends, Appendix 7 contains (for the lossless case) tables that indicate the effect of each circuit parameter on the circuit performance. The mean and standard deviation were chosen as examples, although similar tables could be made for each statistical variable. Appendix 7, Table 5 shows the range over which the parame- ters were varied, and the corresponding circuit variable over which they had the largest effect on its mean value. Table 8 shows the standard deviation of the circuit variable each parameter has the largest effect upon. Here the para- meter range is constant. It should be noted, for different ranges of the parameters, the variable of maximum effect, and the magnitude of the effect, may change. Table 6 shows which variables each circuit parameter can effect a change in the mean value of at least 0.1% or greater. Table 7 15 16 shows which variables each circuit parameter effects the standard deviation by 0.1% or greater. Again, the specific variables affected may vary with range; these tables are provided only as an example. . From Tables 6 and 7, several interesting effects can be noted. The only circuit parameter that affects the mean value of every variable is the quartz rod. The only circuit parameters that affect the standard deviation of every variable are the quartz rod, and the cavity width (b). From Table 8 we note the effects of T81 and T82 are complementary - they invidually tune different aspects of the circuit, as expected. In this circuit, a major source of the loss of power (inefficiency) is loss in the stabilizing load. From Tables 6 and 7, the effect of adding a band stop filter can be seen in the variance of power and efficiency with the changes in the length of the filter (LF) and impedance of the filter (2C). The reason for the many different types of statistical data made available from this program is more obvious upon examination of Tables 5-8. A particular para- meter may increase only the mean, or the standard deviation, or the largest maximum value, etc., and depending on the type of 'curve shaping' that is required for the specific application, the effect of any single parameter could be lost if only one type of statistical analysis was used. A feel for the general response of the oscillator can be obtained from the graphs in Appendix 10, Figures 12-27. Figure 12 shows the reflection coefficient as the 1/4 wave 17 transformer varies 4 thousandths of an inch. The reflection coefficient locus pivots about one point (18.35 GHZ) and expands outward. Figure 13 shows a similar response from varying r1 by 4 thousandths, except the point of rotation has moved up in frequency causing the area of maximum change of reflection coefficient to move up in frequency also. Figure 14 shows the effect of varying the iris (d1) from eliptical to circular. This has a large effect on the low frequency values of the reflection coefficient. The effects of reflections from the stabilizing load can be seen in Figure 15 where Ystab (Y5) varies from 0.6 to 0.0 mmhos. The value of Ystab in the other plots is matched to Yo, which is normally 20 mmhos. The variation of load power (with load matched to output waveguide at all frequencies), caused by varying r1 4 thousandths is shown in Figure 16. The point of maximum power moves down in frequency and the curve becomes more peaked. Figure 17 shows (as the cavity height (a) varies 4 thousands) how the point of maximum power moves up in frequency and the curve becomes more peaked. Figures 18 and 19 show the variation of the imagi— nary part of the input admittance as the air dielectric tip length (Lx) and the quartz rod depth (Q rod) are changed. Lx has its largest effect at the ends of the frequency range, and the quartz rod changes the maximum value of the curve. The curves tend to remain constant over the upper part of the frequency range. Figures 20 and 21 show the effect of Lx and the quartz rod on the real part of the 18 input admittance. Lx affects the ends and upper portion of the curve, and the quartz rod shifts the point of maximum value and spreads the curve out. The curve tends to its highest values at the upper frequencies, and has less variation in this region. The variation in the efficiency with changes in the cavity height and iris is shown in Figures 22 and 23. The movement and increase of the maximum value occur as the coupling of the output load changes with the iris diameter, and a changes the resonant frequency of the cavity. At higher frequencies, where the real part of the input admittance is largest, the efficiency drops off rapidly and approaches a constant value. Coupling to the output is decreasing, and more power is being lost in Ystab' In Figures 24 and 25, QL is seen to drop to a constant value at the higher frequencies, and responds to the quartz rod more than the air dielectric tip. Increasing 0L at the higher frequencies will increase the efficiency and the bandwidth also. Figures 26 and 27 show the input susceptance slope vs. efficiency with the largest changes at the lowest and highest efficiencies, as expected. Validity of Modeling The verification of the results of this program come from several sources. The actual values calculated by the program were compared to values calculated by hand using the defining equations to determine if there were any errors in implementing the equations in the computer model. More importantly, the response of the circuit as predicted by the computer model is identical to the response of actual cir- cuits that were constructed and their performance measured at Bell Labs by Dr. J. Freeman. Also, the computer model behaves 'logically' -- if the stabilizing load is open- circuited and there is no loss in the ciruit, the efficiency becomes 100% because all power input to the circuit goes to the load. 19 Future Enhancements There are several enhancements that may be added to the program to aid in analysis. To prevent an unstable config- uration, it would be desirable to have the program perform sensitivity calculations, perhaps by varying each parameter a small amount and checking the change in the outputs. Also, a small noise generator could be added to the diode generator to see its effect. Another area for further work would be to consider this program as a subprogram to a master control program. Then it would be possible to sequence the variables one at a time, then two at a time, etc., and try all orders of all combinations, and try combining several oscillator models. Due to the amount of computer time this would require and the costs involved, this would almost have to be done under a grant. The use of the Textronix graphics terminals to view the plots would be a great aid in this. It would also be possible to have the program analyze its own data, and return a set of circuit parameters that would optimize (for example) the mean value of the circuit effi- ciency. Another area for further investigation would be further refinements in the ABCD parameter models. It may be possible to use a rectangular aperture for the coupling bet- ween the coaxial line and resonant cavity to improve the bandwidth, for example. To incorporate nonlinear effects, piece-wise linear models could be used. Another possibility 20 21 is to incorporate more experimental data into the ABCD models. This would be particularly useful in the ranges where models are not available. Summary A new computer model for analyzing microwave oscillator circuits has been presented. This model provides an extre- mely accurate simulation of the frequency response of the actual circuits. Since it is extremely easy to analyze the circuit using computer simulation, this model should be invaluable in the design and optimization of microwave cir- cuits. The capabilities of the circuit have been covered and some of the results from the program have been pre- sented. Since the program is written in Fortran, it can be easily used at any computer installation. 22 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (l6) (17) References Microwave Engineer's Handbook, Vol 1, Artech 1971, p. 75. MIT Radiation Laboratory Series Waveguide Handbook, Vol. 10, Editor: Marcuitz, p. 239. MIT Radiation Laboratory Series Waveguide Handbook, Vol 10, p. 368. MIT Radiation Laboratory Series Waveguide Handbook, Vol 10, p. 242. Step Cap Microwave Engineer's Handbook, Vol. 1, p. 111. M.I.T. Radiation Laboratory Series, p. 73. Field Theory of Guided Waves, Collin, McGraw-Hill, 1960, p. 183. Microwave Techniques, Mooijweer, MacMillan, 1971, p. 250. Microwave Devices, John Wiley, 1976, Editors: Howes, Morgan, p. 216. Circuit Design for MM-Wave IMPATT Oscillators, N. Kenyon, M.T.T. Microwave Theory and Techniques, Microwave Symp., June 1970. Equivalent Circuit and Tuning Characteristics of Resonant-CAP Type IMPATT Diode Oscillators, N. Kenyon, Euro. Microwave Conference, 1973. A Study of a Single-Tuned, Waveguide-Coaxial Circuit for IMPATT Oscillators and IMPATT Diodes Characterization, R. Wroblewski, Euro. Microwave Conf., June 1974. Short Communications Single-Tuned Solid-State Microwave Oscillators, N. D. Kenyon, Circuit Theory and Applications, Vol 1, 1973, pp 387-393. 140 GHZ Silicon IMPATT Power Combiner Development, Kai Chang, G. M. Hayashibara and F. Thrower, Microwave Journal, June 1981. Principles of Microwave Circuits, Montgomery, et a1, Dover, 1965, p. 168. MIT Radiation Laboratory Series, Vol 10, p. 364. CAD of Microwave Circuits, GUPTA, GARG, CHADHA, ARTECH, 1981. 23 APPENDICES APPENDIX 1 CIRCUIT AND CONTROL VARIABLES Y3 7 L8 3'i "-Lchoke ”ll ) [r V— BRRaV - g in T81 T82 HEB-WU, ' ~ 2 Id 1" L H 31 b" .31. 4 Figure 1: Side View Li“ 1.: Ls 3?, Ys T31 T82 coax tip inner diameter coax tip outer diameter cavity height tuning screw height output height band reject filter coupling pad height cavity iris diameter dielectric length fichoke BRF length cavity to BRF length step cap. to cavity BRF to Y length air tip Iength output load stabilizing load tuning screw 1 tuning screw 2 of Oscillator coax inner diameter coax outer diameter cavity width tuning screw width output width coupling pad width T82 to YL length cavity length iris to T81 length TS) to T52 length Q Rod quartz rod Figure 2: Top View of Oscillator 24 12 An. pwsosfio pcmam>flscm .m stowam s a we--. sq m ~n--- 5 m.“ m” I L . m ,. a: ...... m _ . on 1:: mm? 3--..“ A m u .N mu H a a... fi - m. x cm“... ::::: o 3 _ _ a .N . - _ .. N .S N . .. _S.~\9H.~\.a ma.- ----. . h m u u H t m: 3 I: m. 0.. m an .23 --..m an -12 Plane 10 11 12 13 14 15 16 17 18 19 A1 26 p3 TABLE 1 PLANES USED FOR CIRCUIT DIVISION Physical Location Plane of diode Edge of transformer (in air) At capacitive step (in teflon) Above step (in air) At physical bottom of cavity Physically at center line of cavity, electri- cally at plane of coupling pad Top of cavity to bottom of band reject filter Band reject filter Outside of coax to quartz rod waveguide end Quartz rod Quartz rod to iris waveguide end Iris Iris to T81 waveguide end Tuning screw #1 T81 to T82 waveguide end Tuning screw #2 T82 to YL waveguide end YL plane Y8 plane Iw(1)=o Iw(2)=o Iw(3)=o Iw(4)=o IW(5)=1 IW(6)=1 IW(7)=1 IW(8)=1 IW(9)=1 IW(10)=1 IW(11)=0 IW(12)=0 IW(13)=0 IW(14)=0 IW(15)=0 Al 27 p4 TABLE 2 PROGRAM CONTROL VARIABLES Print Control Variables No losses in circuit, Print no losses No band reject filter, Print no BRF No tuning screw 1, Print no TSl No tuning screw 2, Print no T82 Print values of frequency, variable value, quartz rod reactance, B array, YE array, BE array, and all ABCD parameters at each frequency, for each value of variable. See Table 9, Appendix 9. Print plane number, current value of variable, Y array, YN array, V array, C array, at each plane, for each variable value. See Table 10, Appendix 9. Print ROM and RCA arrays at each plane, for each variable value. See Table 10, Appendix 9. Print plane number, EA, CMAG, CANG, and PL arrays at each plane, for each variable value. See Table 10, Appendix 9. Print plane number, YC and PC array at each plane, for each variable value. See Table 10, Appendix 9. Print plane number, SSA, QL arrays at each plane, for each variable value. See Table 10, Appendix 9. Plot input reflection coefficient on polar axis. See Figure 12, Appendix 10. Plot real part of input admittance vs. frequency. See Figure 16, Appendix 10. Plot imaginary part of input admittance vs. fre- quency. See Figure 18, Appendix 10. Plot real part of load power vs. frequency. See Figure 20, Appendix 10. Plot efficiency vs. frequency. See Figure 22, Appendix 10. A1 28 p5 TABLE 2 (cont'd) IW(16)=0 Plot QL vs. frequency. See Figure 24, Appendix 10. IW(17)=0 Plot susceptance slope vs. frequency. See Figure 26, Appendix 10. IW(18)=l Print all statistical values for all variables at each frequency, for each variable value. See Table 11, Appendix 9. ' IW(19)=1 Print array IEXT with statistical outputs or plots (for each identification). See Table 11, Appendix 9. IW(20)=1 Print best values of statistical data. See Table 12, Appendix 9. Run Control Variables Values of NA array elements to vary that parameter (order is determined by location in NA array). NA NA NA Value Variable Value Variable Value Variable l Lx 13 a 25 Kc 2 LTR 14 b 26 Aw 3 Lo 15 r1 27 a' 4 LE 16 r2 28 b' 5 2c 17 r3 29 a” 6 LTl 18 r4 30 b' 7 LT2 19 d1 31 ORL depth 8 LT3 20 d2 32 Ystab 9 L5 21 d3 33 YL 10 LC 22 (er)¥2 11 T81 23 CL A 12 T82 24 a D APPENDIX 2 PROGRAM ARRAYS Length Arrays and Typical Values Typical Array Value Array Typical Element Variable (inches) Element Variable Value 8(1) Lx .015 D(l) a .42 8(2) LTR .115 D(2) .17 8(3) L0 .010 0(3) r1 .032 8(4) HC/2 .085 D(4) r2 .0735 8(5) LF .073 D(5) r3 .0315 8(6) LC/2 .225 0(6) r4 .05 8(7) LTl A g'/8(0.ll9) D(7) d1 .26 8(8) LT2 A g'/4(0.239) D(8) d2 .16 8(9) LT3 .l D(9) d3 .145 8(10) L3 .1 D(10) P 1 8(11) LChoke A o/4(0.158) D(11) M 1 8(12) T81 .05 D(12) a' .42 8(13) T82 .05 D(13) b' .17 8(14) ORL 1 TURN D(14) a' .42 D(15) b' .17 N16) A 1 mm )- cav 1 one) A 'c: 1 D(19) (E r)lr’Z 1.435 D(20) l.'G 1 Note: Variables set to "one“ (except QRL) are evaluated elsewhere in the program. 29 A2 30 p2 Variable Arrays Array Variable Array Variable Array Variable G(1) (ya)Lx YE(1) Yod B(l) Ystep 6(2) (YTR) LTR YE(2) Yot 3(2) zC 3(3) (YTR) LTR YE(3) Yot 3(3) Y2 G(4) (Ya) LO ys(4) Yo 3(4) -j32 G(5) (Ya) HC/2 YE(5) Yo 3(5) 21 c(5) (Ya) HC/2 YE(6) Yo 3(6) 23 G(7) (Ya) LF YE(7) Yo 3(7) 21' G(8) (ya) LP YE(8) Yo 3(8) 23' c(9) (yc) LC/2 YE(9) Yo' 3(9) YL G(10) (YC) LC/2 23(10) Yo' 3(10) Ystab G(ll) (Yo) LC/2 YE(11) Yo' 3(11) ‘jBa G(12’ (Ye) LC/2 Y3(12) Yo' 3(12) -ij G(13) (ng') LTl YE(13) Yo” C(14) (YWQ') LT1 23(14) Yo' G(15) (ng') LT2 YE(15) Yo” G(16) (ng') LT2 YE(16) Yo” G(l7) (ng') LT3 23(17) Yo' G(18) (Ya) Ls YE(18) Yo AT(l) “air 33(1) Bair AT(2) “die 33(2) Bdie AT(3) O‘choke 33(3) Bchoke AT(4) “cavity BE(4) Bcav AT(S) owg BE(5) ng AT(G) O("mg BE(6) 3"wg Elements Variables Mean S.D. XP Xsum/Xp Xm Largest Freq. largest Smallest Freq. smallest -AV Device angle Freq. close .95 Freq. close 40 # |3w| 18.7 # BW Mean S.D. xP Xsum/Xp Xm Largest Freq. largest Smallest Freq. smallest -AV 2 3 R.PWR Yin x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 31 8V Array Utilization 4 5 6 MRFL ARFL Yin (s10pe) x x x _x x x x x x x x x x x' x x x x x x x x X X x x x x x x x x x - x x x x x x x APPENDIX 3 TABLE 3 ABCD MATRIX MODELS T-Pad A = l + 21/23 3 = (21, 22 + zzz3 + 2123)/z3 C = 1/23 D = 1 + 22/23 Pi-Pad A = Yz/Y3 B = 1/Y3 c = (Y1Y2 + Y2Y3 + Y1Y3)/Y3 D = 1 + Y2Y3 Series Element A = l B = 2 C = 0 D = l Shunt Element A = 1 B = 0 C = Y D = 1 Thru Connection A = l = 0 C = 0 D = 1 32 Transmission Line Input Admittance Input Current Reflection Coefficient Zero Length Attenutator D > >< U 0 (I! A3 33 p2 TABLE 3 (cont'd-) cosh YL (sinh YL)/Yo Yosinh YL cosh YL C 4' (D)(YL) A + (BHYL) Yin " YL (10x + 10“X) *2 Zo/(lox - io‘x)¥2 (10x - lo-X)/2z0 (1ox + 10"");2 ol/ZO A3 TABLE 4 p3 ABCD Matrix Equations Array Element Section Name ABCD Elements Defining Eqpations 8(1) X) A1 = COSh Yaier Yair = O‘air + Bair Air dielectric Bl = (sinh Yaier)/YOD‘ coax tip C1 = YODSinh Yaier YOD = (601T1(r2/r3))-17Q D1 = cosh Yaier A(2.X) A2 = COSh'YTRLl ‘YTR = O"TR + BTR Teflon 32 = (sinh YTRL1)/YOT YOT =YO/(g;r)92 dielectric coax line A(3,X) step discontinuity A(4,X) air coax A(5,X) coupling pad A(6,X) Top of cavity C2 = YOTSIHD YTRLI D2 = COSh TRL]. YOT = E r/(601n(r2/r3)) A3 = 1 Ystep = f/9.95 x 1012 B3 = 0 C3 = Ystep D3 = 1 A4 = cosh YairLo B4 = (sinh YairLo)/Yo C4 = Yosinh'yairLO D4 = cosh YairLo A5 = 1 ' jbb/(Ystab ' jBa) 35 = (Y5... - jB.)'¥9 C5 = 'jBa = ij ' BaBb/(Ystab ' jBa) D5 = jBa/(Ystab ' jBa) A6 = COSh YairHC/z B6 = (sinh YairHC/2)/yo . HC C6 = YosinhYalr /2 34 A3 35 p4 TABLE 4, continued Array Element Section Name ABCD Elements Defining Equations A(7,X) Cavity to BRF A(8,X) BRF A(9,X) Coupling Pad to Quartz Rod A(10,X) A(11,X) Quartz Rod to Iris A(12,X) Iris = cosh YairLF = (sinh Yair LF)/Yo- = YosinhYair LF = cosh Yair LF = 1 =2 = 0 = 1 cosh‘Ycav LC/2 = (sinh‘Ycav LC/Z) Y0 = Yo' sinh‘Ycav Lc/2 = cosh Ycav Lc/Z - yq = 1 LF = ()‘0/4 ’ Hc/Z) Zc = zchoke tanh‘Yl A o/4 2 x 10'6neper/mil Bchoke = 2"/30 zchoke = 6 = cav + choke 1choke = ochoke = cav cav YOY 04 flr221n(r2/r1) (ab lwg) Y0' = l +Tin " .I< . Yq = cosh Ycav LC/Z = (coshY cav Lc/2)/Yo' = Yo'coshY cav Lc/2 = cosh Ycav Lc/Z = l = 0 = jBZ = l A3 36 p5 TABLE 4, continued Array Element Section Name ABCD Elements Defining_Equations A(13,X) A13 = cosh ng LTl Iris 813 = (Sinh ng LT1)/Yon to T82 C13 = Yo”sinh ng LTl . D13 = cosh ng LTl A(14,X) A14 = 1 + 21/23 21 = 22 = ~jxa TSl 314 = (zlzz + 2223 + 2123)/z3 23 = -ij C14 = 1/23 xa = 20' (1.55x10'3t1") 314 = 1 + Zz/z3 xb = 20" (3.9525/t1") A(15,X) A15 = cosh ng LT2 T81 to B15 = (sinh ng LT2)/Yo' T82 C15 = Yo”sinh ng LT2 D15 = cosh ng LT2 A(16,X) A16 = 1 + 21/23 TS2 316 = (2122 + 22z3 + 2123)/z3 C16 = 1/33 316 = 1 + z2/z3 A(l7,X) A17 = cosh ng LT3 T82 to B17 = (sinh-ng LT3)/YO" YL C17 = Yo'sinh'ywg LT3 D17 = cosh'ywg LT3 A(18,X) A18 = cosh'yair Ls BRF to B13 = sinh'yair LS Ystab C18 = Yosinh-yair LS D13 COSh'Yair Ls A3 37 p6 11 -——D ~——D 12 VI A B V2 + "43 + 11 o o 12 V1 V2 CF—- - Figure 4: ABCD Voltages and Currents QL r E 'JX Jig min 0.12- 22 ——é> o—l LI—H—o -jfb 4J-' 3: 23 1: a'-——>5 0” ' t—0 O A=1+z,/22 3=(z,zz+2223+z,z3)/z3 o=1/’z3 D=1+zz/23 z1=-Jxa zz=z1 z3=-be YA=25(1o55x10'3t) 2.13:2?“ 309525/15) Figure 5: Tuning Screw Model .A3 38 to Y o ‘I‘ L o -ij M to -JBa -jBa Y S C} Figure 6: p7 O Y P 3 =3- 2 a Zlor2 1n(r2/r1) Yvor221n(r2/rl) M to (3- 7.1 d2 3 M=(( 9 )(———) 0.2)d1 /6 d1 Coupling Pad Model W. O T 01— Figure 7: -14 Yd=10.05fx10 Step Capacitor Model A3 39 p8 6-19 Yin 6-1 . 9-18 . (Yin -JBa)(Yin -JBa) Yin: '6-19 9:13 . 'JBa ...___., 'jBa (Yin ~jsa+yin -33b) 9-1 -JBb Yin -jBa I (J Yinj> 1 Figure 8: Coupling Pad Inward Reduction Model 0 Ystab-jBa —()——0 I6 19 + + O——- —__o + A B i [:> V6 on v9 * - - V6 -jBa “393 v9 5 A. .1, :5 Figure 9: Coupling Pad Outward Reduction Model APPENDIX 4 Quartz Rod Analysis In order to obtain a model for a quartz rod in a rec- tangular waveguide (Fig. 10A), we will consider the quartz rod as a variable admittance. We need then to determine (Fig. 10B) Yq, where Yq = Gq + qu ,.,,e/:5""T;3' ‘—“TT’ C%* A .0 and Gq is greater than zero, [ y”,,J T but Bq could have either sign. Figure A Y q From Fig. 10C we note that (> i :0 if we know Pin! and Y0 is Figure B the characteristic admittance C** T 4‘7 T. of the waveguide, in c Yq Y3 Yin""‘"> ‘ then 0- l I Figure C Figure 10: Quartz Rod Model ‘— 1 - Tin ‘- '- — 1'- Ti Yin=m aninn=Yq+1squ+l=T—:—i:—i—2- or Yq --%—}-—%%% - l = -§ +F%gn. By using P in = 01¢ and F in = T real + T imag. whererreal = 0 cos ¢ -2 Yo ((3cos g,+ j(>sin¢) and I‘ imag.= Dsin¢ so an—Ifipcoso-bjpsinct) This is the expression used in the program for Yq. The values for p and o of the input reflection coefficient were obtained from experimental data taken at Bell labs by Dr. J. Freeman. The waveguide was terminated in its characteristic 40 A4 41 p2 admittance and the input reflection coefficient was measured at 20 frequencies from 17.7 to 19.7 GHZ in 0.1 GHZ steps: this was done for 16 turns of the quartz rod, from zero to full penetration. These data values are stored in an array and the proper set of reflection coefficient values are cho- sen depending on the depth of quartz rod chosen by the user. APPENDIX 5 Attenuation, Phase Factor and Impedence Equations For the attentuation in the air dielectric co-axial line -3 p we use (Ref. 6) oair = 2.7 x 10 (f))2(1+r2/r1)/(r2 ln(r2/r1) neper/mil. For r1 = 32 mil. and r2 = 73.5 mil., at 18.7 GHZ “air = 1.99 x 104fi%%223. For losses in the teflon trans- former we must add the dielectric loss to the conductor a )92 mil.) tan5 loss. The dielectric loss is “die = 3.145 figggg where tan5 is the loss tangent (Ref. 7). Experimentally tanG = 0.0028 at 3 GHZ, and tan5 = 0.0053 at 25 GHZ, for our program we use tan 6 = 0.005. Then adie = 1.57 x 102 ( er)1/2/ A neper m. At 18.7 GHZ, “die = 3.57 x 10"5 neper/mil., and the losses in the transformer are: otot a aair + odie = 23.486 x 10’5neper/mi1. For losses in the cavity the unloaded quality factor is given T'X 2 by (Ref. 8) Q0 = _‘*_’_%.I:'.___. = raffl- , and using 4550 as the value for Q0 and L = 450 mil. gives “cav = 1.76 x 10'6 neper/mil. 42 A5 43 p2 The phase factor in the air-dielectric coax is given by Bair = 2 n/Ao. The phase factor in the waveguide is given by so = 2 1T[3&9 whereg Ag is the guide wavelength pkg -.Ao / (1 - (A33)Z)L§ and a is the guide width. The phase factor in the teflon transformer is give by 2w B'r= AT where _ AT =. Ao/( e r)]/2. A5 Method of Impedence Determination In the coaxial line the absolute value of the admit- tance is known as Y0 = (601n(r2/r1))'1. To find the admit- tance in the waveguide, we relate it to the admittance of Y A4 r 2ln(r /r ) the coax by (Ref. 3) Y0' = —9 " 2x9 b 2 1 . This is a consistent with the model used for the coupling pad. This 2 Yo A ab Ag r can be written in the form Y0' = (2 fl r221n( r2 M. The classical expression (Ref. 8, Ref. 13) for the admit- tance of a waveguide is Y0" = X3: ; . This is done by 9 choosing the voltage equal to the line integral of the electric field. If two waveguides of different sizes are joined, the admittance will transfer as Yo" Yoa A Yoa' A ab' A :g_ §;T- - -§Efy; ‘igf‘fT; - arb Ag . However, by choosing the constant of proportionality equal to a/2 between the voltage and line integral of the electric field the admittance becomes Y0" = —%%fir%—— which is the form we use. The admittance between two waveguides then transfers Y " ZY A 2Y a'b'lx'a . as Y , ab Ag ETBTQVT- abyx as in Ref. 16 and the program. 44 APPENDIX 6 Device-Circuit Angle Under steady state oscillation, the IMPATT diode will have a current flowing through it that is a periodic function of time at the frequency of oscillation. Because of the high Q of the cavity, we assume its filtering action will keep the harmonic components of the current small. The current is then given by i(t) = A cos (wt + o ) + (small harmonic components :2 0) where A is the amplitude and ¢ is the phase of the fundamental component. The corresponding device voltage Vd (t) may have in-phase components given by -R(A)A cos ( wt +|¢) and out-phase components give by EKA)A sin( wt + ¢). In addition to the fundamental com- ponents, the device voltage may contain harmonic components which may not be small. The device voltage is Vd(t) = RE( 71A)Aexp[j(wt +¢)])' + (harmonic components) where E (A) RKA) - JYKA) and - fKA) is called the device impedence. Because we are tuning to the fundamental com- ponents of the current and assume small harmonics, the device impedence is a function of A only. The circuit impe- dence as seen by the diode is z(w) = R(w) + jX(w). The voltage developed across this impedence due to i(t) is Vc(t) = R3 ( 2(uDAeXP[j(w t + ¢)]) + (harmonic components). Depending on the magnitude of R(nw) and X (nw), the harmonic components of Vc(t) may not be small. For free-running 45 A6 46 p2 oscillators, the sum of Vc(t) and Vd(t) must equal to zero since no external voltage is applied: Vc(t) + Vd(t) = 0. Substituting for Vc(t) and Vd(t), multiplying by cos (wt +¢) and sin (wt +¢), and integrating over one r.f. cycle, we have [R(w) - R(A)]A 0 and [X(w) + i(A)]A = 0 or, equivalently, [2(w) = i(A)]I = 0 where I = Aexplj3 Figure A oscillation. The i(A) locus the 2(w) locus is called the impedence locus. If, for a Figure 11: Device Angle given frequency, the value on the device line is known, then the required value of the impedence locus for oscillation at that frequency can be determined. This is basically how the program determines the bandwidth of the oscillator. By knowing (from measurements) the device line for a specific packaged IMPATT is at 0.95z1-40° for oscillation at 18.7 GHZ, the program finds the points on the impedence locus which will allow oscillation. By allowing both A and d>to be slowing varying functions of time, it can be shown (Ref. 9) the condition for the intersection between the device line and impedence locus to represent a stable operating A6 47 p3 point is AO|_2%%321: :._3;é£21| sin 9 >0 where 9 is defined in Fig. A. In other words, the intersecting angle measured clockwise from the device line to the impedence locus must be less than 180° for stable operation. The program determines this angle as follows: A straight line is defined through the two points on the impedence locus closest to the desired point on the device line as shown in Fig. 118. Using the xfix 20”) fl . . _b(wg) - b(w1) angles in Fig. B, tanu’ 9(w2) _ g(w1) since 9 = u-u2then w = n‘- 6 and tanUJ= tan (n-e) (A) or tan ¢)= -tan9 so 6 = tan"1 (b(w2) - b(w1)) s—D R Figure 11: It can also be shown (Ref. 10) that the Device Angle power spectrum of the amplitude fluctuation is given by 482(wo)2- 2 I“ (t) 2 = : w a”). 2 4 __ 3X(w°) - “(($37 2 w a—Jg‘fiw ) +(SR(A9) 3w " R(Ao)—Fw—9—) where s is the saturation factor of the negative resistance -A 351%) and is given by s = §(A ) 55A and r is the satura- ~o tion factor of the device reactance and is given by A, a'iuo) HMO) 3A ° H II The power spectrum of the frequency fluctuation of the oscillator is A6 p4 48 When the intersecting angle between the device line and the _ 8X(ub) _ 33(un) impedence locus becomes sharp, 63(00) 'TET‘ ’PR(A°) aw in the denominators of the power spectrum of the amplitude and frequency flucutations becomes small since it is propor- tional to the sine of the intersecting angle. Then, the low frequency components of both the amplitudue and frequency fluctuations become large. The oscillator becomes noisy as the intersecting angle approaches 0° or 180°. The desired angle then is 90°. The program finds the frequency where the device angle is closest to 90° to aid in attempts at noise reduction. APPENDIX 7 VARIABLE EFFECTS TABLE 5 Variables That Parameter has Dominant Effect On (Mean Value) Affected Circuit Variable (in percent) Para- Range meter Eff P Yinr Yini MR AR RS QL Qo Qex (in inches) LX 106 017-01716 LTR 5 LF 6 01-014 2e 3 .01-4.01 LTl 08 011-0126 LT2 6.0 015-019 QROd 359 0-4 Ls .204-.2808 T52. ' 12 .001-.04 a 3 041-0418 d1 12 02-028 d2 .7 0155-0159 iris 11.43 .149-.158 er 2.3 1.44-1.456 49 A7 50 p2 TABLE 6 Variable That Parameter Effects by at Least 0.1% (Mean Value) Para- meter Eff P Yinr Yini MR AR RS QL 00 Qex Lx x x x x x x LTR x x x x LO x x x x LF x x x x x x x x zc x x x x LT1 x x x LT2 x x x QRod x x x x x x x x x x Ls Lc TSl x x T82 x x x a x x x b x x x x x x x x r1 x x x x x x r2 x x x x r3 x x x x x x x r4 x x d1 x x x d2 x x iris(d3) x x x at x x x x x x A7 51 p3 TABLE 7 Variable That Parameter Effects by at Least 0.1% (Standard Deviation) Para- meter Eff P Yinr Yini MR AR RS QL 00 Qex Lx x x x x x x LTR x x x LO x x x x x x LF x x x x x x x x zc x x x x x x x x LT1 x x x x x x x x LT2 x x x x x x x x QRod x x x x x x x x x x Ls Lc TSl x x T82 x x x x x x x a x x b x x x x x x x x x x r1 x x x x x x x x x r2 r3 x x x r4 x x x x x x d1 x x d2 iris(d3) x x er X X X X X A7 52 p4 TABLE 8 Variable That Parameter Has Dominant Effect On (Standard Deviation) Para- meter Lx LTR L0 LF r1 r2 r3 r4 dz iris Er Eff P 2.02 32 6.9 Yinr 3.1 .638 yini 14.6 MR AR RS QL Qo Qex .4 .99 6 S65 10 9 11 4.7 97 357 .77 Line # 1 2-12 12-14 15-17 18-165 166-167 168-169 170-171 172-173 174 175-183 189 191-195 196-230 231-242 243 244 245 246-250 251-252 253-255 256-728 Appendix 8 PROGRAM DESCRIPTION Specify input and output devices Dimension arrays Specify complex variables Define functions used in Program - cosh, sinh and U (reduction function) Fill RHOL and PHIL with all quartz rod magnitude and angle values Fill D array with default values Fill S array with default values Fill ISTAT with character variables to print with statistical values Fill MEXT with blanks - if losses, Q Rod, BRF, T81, T82 are removed later, array is changed Fill ISTAY with values to print with best statistical values Fill IDASH array with characters to separate statistical values Fill IEXT with run identification number Initiatlize NA, IW, W arrays (W is no longer used elsewhere in program) Choose variables to run and order by setting values of NA ARRAY: 999 terminals run Choose waits, plots, writes, and losses, 0 Rod, BRF, T81, TS2 Initialize loop counter (one loop per variable) Top of loop for each new parameter: kv = current parameter # Set default value of choke impedence Remove losses, BRF, T81, T82, add in 0 Rod Check if done Find desired parameter and go initialize it These lines contain the identification and range variables for each parameter. The ones used for every parameter are: CV = current value of parameter: VI = variable increment; IBCE = name of variable to print: IC = length of variable name (up to 16): IBCD = variable name with equal sign for plotting with dashed lines IL = length of IBCD (up to 16) LN = number of times to increment CV (can have any value up to 10 since is decremented and checked to zero) CVS = starting value of parameter (for printing) CVF = default stop value (for printing) IN = number of decimal places to print on plots 53 Line # In addition, A8 54 p2 FV = value to reset parameter to after all done LK, NR = copies of loop number (LN) LC, LF, ZC Set IBPF to l (to flag band stop filler in use) TS1, LTl sets ITSA to 1 flag tuning screw 1 being used T82, LT2 sets ITSB to 1 flag tuning screw 2 being used AA, AD, AC, AW sets ITEN to 1 0 Rod turn 19 - turn 0 (no zero array index) 729-732 733 734 737-738 739 740-714 742 743-745 746-814 815 816-819 820-827 828-830 831-836 837-845 846-851 852-869 870-899 900-922 923-928 Set losses, tuning screws and band stop filter flags if required for all parameters (i.e., IW = 1) or flag set during initialization (i.e., ITEN = 1) Save copy of CVF Set values of attenuation for air, dielectric, cavity, waveguide Set constant Pi Top of frequency loop for each increment of a para- meter Reset loop frequency counter (LF) and starting fre- quency (F) Top of loop for each frequency increment Jump to set value of parameter being varied Depending on which parameter is being varied, one variable is set equal to CV each increment of CV. Lines 807 and 808 are example of how the Q Rod (for example) can have non-sequential (random) steps and be removed on last increment. Store frequencies run in printing array Calculate circuit values that have size or frequency dependence Find 0 Rod turn desired ( = JJ). Fill RHO and PHI with magnitude and angle values for that turn from RHOL and PHIL: convert PHI to radians Calculate waveguide wavelengths Calculate attentuation used with each of the 6 dif- ferent phase factors If no losses desired, reset attenuation array to zero Calculate the 6 different phase factors Calculate characteristic admittance for each of the 18 ABCD matrices Calculate the 7 admittances that shunt transmission line sections. Lines 890—893 checks that we don't set YL Y ’ stab, Yo“ or Yo if they are being varied. Calculate gamma ( +j ) for each of the 18 ABCD matrices Pretend all ABCD's are transmission lines and ini- tialize them Line # 929-977 978-993 994-995 996-1016 1017-1020 1021-1029 1030-1038 1039-1042 1043-1055 1056-1075 1076-1086 1087-1090 1091-1105 1106-1132 1133-1163 1064-1245 1165 1166 1167-1174 1175-1180 1181-1186 1187-1191 A8 55 p3 Properly initialize ABCD matrices of shunt elements and tuning screws, etc., and remove BPF, 0 Rod, and T81, T82 if not desired Calculations of input admittance, done in three sections: YL to coupling pad, Ystab to coupling pad, coupling pad to diode. YL (Y(18)) must be initialized (line 978). Indexes J and K are reversed because we are counting down. Lines 987 and 988 are coupling pad reduction. Find input current from diode Find volatage and current at each plane. Lines 1005-1010 are coupling pad outward reduction. Normalize input admittance. Find reflection coefficient at each plane and con- vert to polar. Check real part of rectangular reflection coefficient to avoid dividing by zero. If zero, set angle = 999.99. All calculations for this frequency interval are done. Writes are now performed of variables that will change with frequency. As with all writes in this program, a check of the flag array IW is performed to determine if the write is desired or not. Increment loop count and frequency value and check if done. All calculations for this increment of this para- meter are done. Writes (if desired) are now per- formed for the variables over the entire frequency range. Now calculate load power and efficiency and input reflection coefficient for each value of fre- quency (these all have 20 values - one for each frequency) Calculate QL, RSA, QB, 00 - these have 19 values since they depend on frequency differences. Increment variables and loop if more increments desired. Take lst values of arrays for plotting. Write rest of data for parameter. Initialize arrays and variables for statistical analysis. Values set to 1000 are beyond possible values (so find smallest). 20 point variables statistical analysis. Set up for mean. Set up for standard deviation (temporary storage). Find 6 pts. larger and smaller than reference and their average value. Value closest to .95 (mag. rfl. coeff.) Value closest to -40 (angle rfl. coeff.) Device window magnitude. Line # 1192-1196 1197-1207 1208 1209 1210 1211 1212 1312 1214 1215-1216 1216-1218 1219-1220 1221 1222 1223 1224-1226 1227-1233 1234 1235-1240 1241 1242 1243 1246-1321 1322-1329 1330 1333-1336 1338-1340 1341-1142 1343 1344-1349 1350-1358 1359-1365 1366-1376 1377-1385 1386-1391 1392-1452 1453-1465 1466-1593 1494-2073 1594-1595 1596 1597 A8 56 p4 Device window angle. Largest and smallest value over parameter range. Find SD Find mean. # greater. # less. Check to avoid division by zero. + average Jump to avoid resetting. If zero divisor set + average to zero. Locate largest and smallest. Bandwidth Carry over Avoid zero divisor Average Set to zero if divisor = 0 Device angle Largest value Change zero to reference #. Frequency of largest Value of smallest Frequency of smallest 19 point statistical values using same format and variables as 20 point analysis described above. Initialize for finding best values of statistical variables - VAKP = Varaible values; PKEP = best %; AKEP = best value: IKEP = increment # of best value. Count from 1 not 0. Initialize variables beyond range. Skip for reflection coefficient. Find largest statistical value. Save largest as reference. Largest valid data. Find value furthest (less) from reference (not largest of small) Find smallest S.D. Smallest average value of # of points less than reference (skip if reference) Translate numbers if reference is best value. Get rid of non valid numbers. Find best statistical values for 19 point arrays, same format and variables as described above for 20 point arrays. Write circuit configuration if desired. Statistical writes (if desired). These are the plotting routines. All follow the same format so only one is described in detail. Check of plot desired. Set buffer size Set plot limit Line # 1598 1599-1602 1603-1606 1607-1626 1627-1629 1630-1643 1644-1655 1656-1670 1594-1670 1671-1737 1738-1804 1805-1871 1872-1938 1939-2006 2007-2073 2074-2076 2077-2141 2142-2144 A8 57 p5 Set new origin Write titles Draw lines under titles Write no losses, etc., depending on circuit con- figuration. If one title is missing, others will automatically move up. Write run # if desired. Write variable name with equal sign, value of variable and draw dashed line to that curve (twice). Draws circle, plots angle values and marks. routines draw axis after PLIMIT routine and scale to scale plot values to fit on graph. Plotting routines that plot multiple curves on same graph. Reflection coefficient plot. Y real vs. Freq. Y mag vs. Freq. Real pwr vs. freq. Eff vs. freq. QL vs. freq. SS vs. freq. Find variable just run. Reset variable (any special run specific (losses) etc. can be reset now) Increment counter check if done, Other call conditions if not, loop. 58 APPENDIX 9 .e eeouoouon.u «ouuonouu. .o «commuuoc. .a .o eoouoouen.u .e oe.uenecu.u «e-uncuno. .o ne.ueo~eu. .o .e oe.uenoo~.n .a ”someone". .a .o .a .e .e ue.ueeoo~. .e eoouoeouo. «cououopo. .o «someonco. .e .o ooouoooae. .o "o.ueeee~. .o .e .o .o .o «eoueeoau. .e oo.uonnpp. «anuoeooo. .o «eouunuoo. .o .o oaouonnpp. .e ueoueeeou. economoou.u .. .e .e .o «censuses. .e oe.ucueo~. ~euunp~eo. .e neouoeuon. .o .o ao.uo~ao~. .e we.ueoee~. co-unuooc.u na-uo~nnu. .o .o .a nooueeeeu. .o eoououeaw. «auunpuom. .a noouoouou. .o .o oo.uo~oo~. .e neouoeoou. .. .e .e .e .o «e.ueeoou. .e seemenoon. «onuopoau. .o «ooun~oon. .e .o ae.uenoon. .o oo.u~o~oo. neouooopu. .e «concouun. .o .o aoouuowoo. .e ooounnuoo. «unmanned. .o «concouun. .o .e ooouacuao. .e eeoucooeo. «oouuonou. .o “couch“... .o .o oeouoocoo. .o «someoeou. «ouuoopsu. .o .o .e .a «eeueeeeu. .e .o-uo«u-.u «enumepnn. .e «commonau. .e .e nonucn«-.u .o ooeucoaom. «ouuponun. .o «oouueonn. .a .- scone-cam. xuxe nouoewumm m manna 51911E-02 0971035‘02 .OIBSEE-OZ 0358935'02 .29780E-02 .25529E-02 0325365-02 .89608E-02 .298805-01 0938885'01 .346275-01 0266175'01 .218505-01 .18935E-01 .170205-01 .1561TE-01 019683E°°1 0139235-01 .13324E-01 oIZBSQE-Ol LP V 0104825‘01 011093C'01 0127765’01 .19997E'01 0169145’01 0206135’01 0266075‘01 0365165'01 0913995'01 019°Q9E’01 046110E’02 0160005’02 0159955'02 021805E'02 0282955‘02 0300525'02 0900915'02 oQSIQBE'Oz 0.96515’02 0538965'02 RCH 59 TABLE 10 VARIABLE VALUES 0965715‘01 087620E’03 0178695'01 .66679E'01 OSSQOOE'OI 0.79925’01 0605275’01 0166735900 0555865900 .81646E000 0699175000 0995175‘00 0906.15.00 .02982275000 .352255000 .316635000 o29163£*00 .21315E900 .259008900 .297875009 0238765000 RCA CV YN .19500E900 .213806000 .237672000 .26968E000 QSIQGQE‘OO .38347E000 0494976900 .67931E000 .77014E000 .35438E000 eOSTBOE-Ol .30583E-01 .29756E-01 .405635-01 .525455-01 06.0915'91 01.5825‘01 .839895-01 .92311E-01 .10011E000 olSOOOOOE-Ol '0228205001 -0216835901 -.209245001 ‘0193005001 “0173555001 'ol53705‘01 '0130055001 -0102185001 “0756595990 ’0‘2605C‘00 “0192595900 '0101975991 ’0129035901 '01077BEOOI ’0158895901 '016986E’01 -0167615901 -016837E001 -0167935091 ’01667‘5‘03 01517366E603 08954214E*00 .86275985000 080298922000 090009245000 .9205038E000 .90732695000 07963427E000 05121023E¢00 oZlSSQlOEOOO .22231335900 03381919E900 .42243445000 0.19131IE.°° oSZOlOQQEOOO 059999215000 .51291905900 .59101265000 06056015E900 06178009E000 01556910E903 01531127E903 .14969Q9E003 .10997635003 013796365903 012719735903 .11060296003 .93636695002 .10634146003 .1630597E003 01753615E003 017591805003 017469855003 .17331785903 01719899E003 .17078885003 016971685903 01687692E003 016788055003 V '0521525900 '002028E900 -0326895900 “0239305900I -OIG‘ZSE.OO‘ '010978E000 '0999715'01 ‘0173325000 “0387735000 -0751615900 -0113155991 '0133295901 -0139695001 “0126.05.01 '0115615001 -010989E.01 “0952605900 -0868955900 '079718EOOC '0730515000 '0637975-02 -.5382¢E-02 -0937365'02 -033011E’02 ’023FB9E‘02 '0166095’02 ‘0157155’02 ’0282715002 -0655555'02 '0131585’81 ‘0209935'01 '0299505'01 '0260375‘01 -0252225'01 '023779E'01 -0222315’01 '02C1915'91 '019519E'91 '018917E'01 -017995E'01 A9 p2 C -.26628£-01 -o26898F-01 -0279615’91 -.28¢01£-01 -o29835E-01 -0319635'01 -0399285-01 -0388655-01 '092907E-01 ‘09‘913E'01 -o92606E-01 -.37159E-01 '031999E'91 -o21164E-01 -629165E-01 -.22123£-01 -.20701E-01 '0197006-01 -o18961E-01 -.189OSE-01 A9 PB 60 nolmhooommwo nonuncnmmhwo nuawnaaomcno nocuhuwwhnCo .neuumeenepm. nonwhnncnnho u::wcnuw~nue Nolmmhfltwmdo Nulmccmwmumo Nonwmnnaoswo «aluquRmeo uuowmhhwnmmo «almssmmomno untmncamntao maluonmquuo noluammuoww. nnlutmmohcoo nonunmaohume moausowcnmc. nnluncsonno «uJoa ace; «counomnmnoeo «counconccCon woounappowoou «communwwuoou «oouoowhemno: «cowooowohnos «oouonoaonnon «oouocmhhnnol «oouaoooswnon Nooucomoomnoo «counuhccoool «counJhOQNOol «oouoncmnooen Noouoomuomool «oouomounwnon «oouooonncneu Noouopmoowcen Noouoowoomcol moounpunmmnon «commowcohnol ¢zuo u .uuauruooauov-announvnuflnn-unuuuuuuovounuunun-unava-upon-uupaw.unanuununuunnuauuuunuuuDun-auunuuuuuuuuuun «bumeaxac ~osnwsve a Quad. 0» coca. an n coca. a 04 ¢ can«nun-vooauouunauuuuou.u-loo-annouwouuunu.uuuunau-u-uu-nunuuvuuuuuunoan«unanno-uuupucanon-nonuuu-ununnu mm» a, an» n: 580 or mummoa ox unannouuuoannouuuouun-uua-unouuuuuuuouuuuu-ucan.oo-up-uuuun¢upuuuuu-nouncon-nua¢poua¢onuopouuuonuununuuu meeee> Heoeemaempm F a @493. 63 TABLE 11 (cont'do) 818$38888811338118!301813883080!881$13888801108808108183088911808 01.10E911 0111. “7006 55092 32009 .10205011 .57326-01 16.23 16.12 156.1 .1030EO11-.71b05-02 ‘1.527 '1. 552 ‘19.17 .10905911 '.57OQE-01 “6.616 '9. 321 -162.3 863368851831133988!8113313363883188$1883883133331883806886818'813 .10506011 ‘o0176E°01 “10.79 '11.25 -262.7 .10606011 -.6020£-01 -9.761 ‘10.10 '306.9 .10706011 .671‘E-I1 '0.016 '6.227 '315.0 .10306011 -.~969E- '01 “6.123 '6.151'290.0 63818833888318833813888383188331681333888311933113183833888181388 .16905011'.33226-01 'B.177 -h.251 '291.5 .19005011 '.2.995-I1 '2.766 '2.007 “107.7 .19105011 -.1169E-01 -1.622 '1.653 ‘127.3 .19205011 '.9673E-02 -.65B6 '.6620 '50.60 831883118883883108838311881883636386133183383861881838191808.1108 .19305011 .27Q7E'03 .h23ZE'01 .6275E’01 “.257 .19506011 .37906-02 .6151 .6196 65.60 .19506911 .59755-02 1.210 1.016 123.9 8088801088118318119100888183088808831988933998S100809188810081883 88818181988180088813088180880881833813888111883810801003808101888 "El! .1730E-l1 17.33 20.57 50.63 870 DEV .79015-01 35.96 62.02 337.6 6 P75 0. 0. 0. 0. O ‘7 .0 .09 .0 '0 - PTS 19. 00 .00 19. 00 19.00 I!!!8885901818888009038081088883888811880839388110108808880880888 LARGESY .1607 96.26 115.6 706.7 F LIRGE .1790EO11 .17902011 .17905011 .1770EO11 SNDLL ‘..17‘E"1 -10079 “11025 -3150. I SHALL .10502011 .10502011 .10505011 ..1070EO11 0 ' ‘7 0. .0 o o 00088880880810808838980018888888088888883198813319800001800808008 A9 A9 64 97 TABLE 12 Best Statistical Values 555555555555555555:5555:5555555555555555555535555555555555555555555555555555555555555555555555555555555555555550555555555550500! IO 105375 IR I0 7“ N0 332 2011. 3231 333333333333333333333333313533333333333333333:3333333333ll3305333333333333333303333335333333933333333333333333333333333333333331 L0 ili'INS FROI .1630 70 H0 09/ 23102 5657 VALUE ORCER-DLRLEMV.INCR.VAE VALUE.0ESY VALUE IN. .1. 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C 1555 5. 5. 5 5 5 5 5 5 5 5 5 1555 5 l .1555 .1555 .1555 .1555 .1555 .1551 .1550 .1555 .1555 3555 .1510 .1535 5 25.00 25.50. 25.500 25.55 15.55 15.55 55.55 15.55 15.55 1555 20. 55555555555555555565555555555555555535555:555553555553555555535355533353555555355555555555555555555555555555)55555555555555555 155555! 5. 5. 3. 5. 5. 5. 5. 5. 1555 5. .5. 1 5 5 5 5 5 5 5 5 5 XHAG 5 5 5 .1395 .1555 .1555 .1555 .1555 .1550 .1500 .1500 .1500 1555 .1515 .15 05 5 .1765 .5;325’52 .75356'51 .5557 '27.55 .1557 55.25 115.5 755.7 1555 ' 15 ..52332‘ ‘01 55:555555555555:555355535555:3555:55355553555553555555555555515555555555555555155555555355555555555555555555555555555555555555 .. 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APPENDIX 1 0 PLOTS REFLL COEFF. RS LTR VQRIES N0 LOSSES N0 BRF N0 T5] N9 T52 LTR : 0-119 90 / ,LTR : 0.115 Figure 12: Reflection Coefficient as LTR Varies 65 A10 66 p2 REFLg COEFF. as RI‘VRRIES ‘ NO LOSSES N3 BRF N8 VS] NO T52 '31 = 0.066 90 ,3: 0.052 '80 Figure 13: Reflection Coefficient as r1 Varies A10 67 93 REFL. COEFF‘ 98 D] VRRIES NO LOSSES N0 BRF N0 T5] N0 T52 RUN1.SET1 DI : 0.403 ,D: = 0.200 Figure 14: Reflection Coefficient as 61 Varies 68 REFL. COEFF. 98 Y8 VRRIE8 NO LOSSES N0 T3] N0 T82 RUNI.SET1 90 (3 (3 L3 0 I I I I I I I I I I I I I ’ ‘ I I . YS ’ I I a I v’ I 4' I f I a a \\\ ‘ x V\\ l”' /' 1 I I ‘ ‘—_"d#/1 l l 4“? -1 0 -O 7 -0 5 -0 2 0.5 G 2 0 5 O 7 1-0 [“190 Figure 15: Reflection Coefficient as Y3 Varies 69 A10 PS ____________________ R1 2 0.0032 8““‘~-Rl : 0.036 FREQ ¥10 Figure 16: Real Power as r1 Varies 2750.00 {300.00 1040.00 xbao.oo 83520.00 1360.00 A10 70 p6 R PNR. V8.F 00.4 1 R : 0.424 .00 neooo :30000 £340.00 1550.00 £20.00 t"!60.00 FREQ. :10 Figure 17: Real Power as a Varies A10 P? 71 Y IMF-1f} V8. F ______.____,_______.____ 05 LX VRRIES N0 LOSSES N0 BRF NO T81 N0 T82 -0.02 .04 -0 -0 A) 1: LX - O \J V ‘ . ‘ I ” I, ’f (D ’I ‘, ' " f ’l ' o i "’ I ’I ‘ I " . O ’0 'I Y IMRG . ‘$§§- /’ -0.08 O\ /;/ ..................................................... LX : 0.0;9 . ":1 ‘u’i {if _—- -l f k a Ban ' x ~~ we {040..“ 1§R:.0? 1320 9 11 n160.:3 l8 'FfiiEa ‘ C“ Figure 18: Y Imag. as Lx Varies Y IMRG A10 72 p3 Y IM9G V8. F 98 OROD V9RIE8 NU LOSSES N0 BRF N0 T31 NU T82 N 9 3 c}- : 1.000 0: o a- as o c}. 2 é_ : 5.000 E 1760.00 1000.00 1040.00 1b80.00 1020.00 1060.00 FREQ :10 Figure 191 Y Imag. as Q Rod Varies 73 Y RE9L V8. F 98 LX V9RIE8 NC LOSSES NO BRF NO T81 NC T82 CO _.' o __ -,. I. - _ l _ - ' - --. “1 750,11. 1801: e - 134.01.; 1050.112. l92c‘ .- .«10 tha Figure 201 Y Real as Lx Varies 4‘ . “360 A10 P9 74 p10 Y RE9L V8. F 98 GRUB V9RIE8 II C) L7 L) U“. ,,0R00 -QRUD I I fl 0 . 1 1 r j ‘ “1750.3, 1300.1: 1040.1: 1960.2: 1920.: 1960.1: FREQ I'D Figure 211 Y Real as Q Rod Varies A10 75 p11 EFF. V8. F 98 9 V9RIE8 NO LOSSES NO BRF NO T81 NO T82 O-41O O-418 1750.00 1800.00 1T84O.OO 1%50-00 €920.00 1950100 FREO XIO Figure 221 Eff. as a Varies 76 EFF. V8. F a 9g 01 V9RIE§ NO LOSSES NO BRF NO TS] NO T82 RUN1.SET1 15 LC 1u~ Figure 231 Eff. as d1 Varies ,01 0 h) C) .4fi 77 NO LOOSE NO BRF N“ TS] NO T82 *0 {'1‘ I ‘,.I .11. 1760. L1 (.3 FREQ «10 Figure 241 QL as Lx Varies 1500.33 1040.30 I;BO.CS 1520. (N J H U 1560.33 913 0.015 O-OI9 78 F OL V8. OROD V9RIE8 98 BRF NO T81 NO T82 0 C) L) L) (J. rd r3 DJ 0 R 0 flu no no 196 1920.03 '37. Op 3 o Fj~8ure 251QL as Q Rod Varies MO 79 p15 85 vs. EFF 05 IRIS V9RIES NO LOSSES NO BRF NO T5] N0 T82 0.24 --nn-mn-un.nu-. ....... JTQIS [1.145 ‘ ‘ s Q ~‘ ~ . Q Q ~ ‘ Q \ s Q . § 5 5 § § ~ ‘ § ~ \ ~ \ 5 § s ~ \ ‘ \ s 5 ~ ‘ § § ~ § § ‘ § § Q \ ~ ~ y s ‘ s s s \ ~ s s s s ‘ Q s . 0.00 0120 0140 0150 0580 1.00 EFF. Figure 261 SS as Q Rod Varies 80 p16 88 VS. EFF 98 OROD V9RIE8 NO LOSSES NO BRF NO T81 NO T82 01124 1.000 5.000 '0 01 0109 0i17 0125 0133 0.41 EFF. Figure 271 88 as Q Rod Varies APPENDIX 1) PROGRAM LISTING c:uput.ourpur:/soo.tnhr 1=xupuv.vapc 2:001901 CANBC20010)0 o 8 l t V 5 6R121905K ‘C‘(8181A .L’GP131100 31.1K(I3201 901050.15 122179C91 Z‘ZOOOROO 1N131)s.155 -10009111 BLE911OH‘ 'HR1912TP ,CF10LO'K GOIO2EBII ‘1)0‘010V TOOOCI(59 A111Y’H11 LoGgPCgToVoYoYCoHCoH80HT'U029IT.UU.VV.UUQX19CZQCo PP‘C)O()OUEHB Ié;0YO1HOK¥;'H...VI=DQ 1UN121012110‘C55V023 0001019909) Yoe'Hoo H:12E100)2)§X900UR00 ‘65! 00220 05.2..." : :U( 0 111.4170. 9 1110304! 0 ’(838‘ 9 ’CUBQ‘ O {In-083C O ’(2617‘ Q 9 RUNN)9O‘29(YLC))O 919 809 358 879 236 736 395 68:1018L(1TAPXZZTA139A015‘229Ie09‘898A006A228‘e00‘8501000‘905A0000A‘16‘oooe“ OEHL2t1OICITHO((TT232T6oeT121T1SoTOOOT77IT21179391436T904T566T119QT099T0169I3 RDIACE‘HHNES ONCSCAOOOA076100013051000A17610001447l000‘056‘000l5556‘1OO|5555A1 PODCDOVRPYHICYHHUD-0.01970.000117000.02160000022200000222000.022220900022220. 112 112 112 112 112 112 112 112 112 212 112 1123 112 1123 1 1 QOQOOOO‘O 10 o.009Q9o01220o01520001860 VVOUUOXX) ( 8 .0368.00367090361000350000335o.0318. =1.2011161.100.139.5590.124.5911.11¢.231. 5 o I 7 15 20 003320.18.53260 8 I o 922 3 O O .02600002630002599002080.°2330.0216. 1120)]222o107o207.2782018Q07225015607893! 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V o 8 Os 6 T. o O Q T. o o Q T. .U .p o}. 80L : V n. .l = V C . .1901 = V P. 20‘ = V 3.00 8 : V n 5. z .2 80 = H 3 o 3 H 3 o = 30° : H 3 9 31° : H 3 O 37 :1" 0 o 31.3: H 1012 1 VV 1. 553 1 VV 3 1 0.033 1 VV 1 50,1 1 VV 1301 1 VV 0.0).}. o 00: C CC NN:000: C CC 1NN:0 o 02: C CC NN :0 o 03 = C CC NNO 090.. C CC ”NC 5 .5: 0011:2055: =3LLFT 0 0E3065.. :31LL‘T00153065::3LL8700151005: :3LLT01 051005: :1.LLT, 3.1.... ..:(c..c: :SF::..D. .. =c..c:=sr=1: :s ..=CC:C:=SF= 23$ : :Cc:c: 35F: 2 : ..l\..c=c::sr 3.... ::(c VIVBCBLNVVNKRBOVvIBCBLNVVNVKR1°V1VBCBLNVVNKRTOVQJVBCBLNVVNKROVV1BCBLNVVNKP 0V1VB CVF1111LCC1LNIGCVIIIILCCIFLNIGCVFIIIILCCILN15CVF1111LCC1LNGCFV111LCCILP. GCVF1 1 2 3 Q l.— 6 1 1 1 1 1 1 5 5 5 5 5 5 P.‘ 390 395 400 405 41:: 415 420 425 433 435 440 445 455 455 g. A1 1 P7 8? 1 1 \I 1 \l 1 1 1 1 1 1 1 : . : . = . = o = o : . N 5 N 5 N N N N 1 L 3 2 L Q 3 L Q L 1 L 2 L R C 7 R C 3 R C R C 5 D C 5 D C H o 0 H a 0 H a H o 2 H c 1 H o 5 1 059 02 5 f. 954 03 5 1 0 Q 5 T. V 0 01 5 T. 0 o 2 5 7. 3 V .5909“ : V .J5OOR : V 0.569“ = V c 9. OD : V a 50 D : V 1 O 360:“ 1 0 550 ..H 1 O 550" 1 O 8 3 2:" 1 O 35:1H 1 0 1 VV 0012 1 V V 0012 1 VV 002 1 VV 1 012 1 VV 1102 1 VV C CC NNO o 06: C CNNO o 01.: CC NNO o o: C CC 8NN02 09.. CC NNO 000: C CC 2055: 2 3LL9I°°152055: 3: LLTOO1£65D¢F : :1.LL.IflJfl-En‘055 : 2 51LLT. 001E555: :3LL'02 or: 1.0.... .: :3 =C: =SF::: ::CC=C:=S:F:= .. :CC..C::SF::: : :C:C: :SF:C:: ..=CC:C= 25F: :: =C:C:C:=SF: CB LN VVNKROVIVBCBLNVNVKROVIVBCBLNVVNKROVvLBCBLNVVN VKROV1VBCBLNVVNKROVV1BCBLNVVN 11LCC1LN5CVF111LC1CLNGCVFIIII LCCILNGCV111LCC1FLN5CVF11LCC1LN5CFV1111LCCT. 7 5 9 0 1 1 1 1 2 2 5 5 5 5 5 Q65 470 Q75 Q80 485 490 495 500 505 510 515 520 525 530 535 A1] 88 = 1 1 1 1 1 1 1 1 1 1 S . : . : o = . .. . 1 N N N N N R L 5 P L A L D L C L S 1 C 5 E C A C A C A C 1 H a Q H c H o H o H o 0 R 7 T. V 0 o 5 T. 0 A 5 1 V 0 D 5 T. V 0 C 5 T. V 0 0511 : V C .J 41 = V C A : V C a. A .. V C 0. 1A : V C 0 1 300H 1 0 2 3.0: 1 O 5 H 1 O = 3 1H 1 O = 320" 1 O : 320 105 1 VV 1 Q01 1 VV 12 1 VV 1 1 302 1 VV 1 1 002 1 VV 1 1 00 NNO o o: C CC 1NNO o 02 C CC NN020: C CC 3NN=000= C CC QNN :000: C CC 5NN:000 LLCCCEQDSKJ : =32LLT1022055 : : 3LLT 0 052055.. :32LLNT 0 052055 = : 32LLNT 0 0.9.2051... :1.2LLNT o o :: ::C:C::SF=C:: ::C:C::5F::: ::C:C::SF:C::E ::C:C::$F:C::E :..C:C::$F:C::.L :: KROVIBCBLNVVNVKR0V1VCBLNVVNKROVIBCBLNVVNVKRTOV1BCBLNVVNVKRTOVYABCBLNVVNV KRTOV... LNGCVIIIILCCIFLNGCVF11LCC1LNGCV1111LCC1FLNIGCV111LCC1FLN15CV1111LCC1F LNIGCV 2 3 5 5 5 1 2 2 2 2 2 2 5 5 5 5 5 5 5M) 545 550 555 560 565 570 7 580 585 590 595 600 605 610 615 A11. P9 89 1 1 1 1 1 .- 1 1 1 1 1 : . 8 o = . = o : c D N N N N N 2 H L 1 L M L 2 L 2 L R A C C C C a C C o o H a m o H o M a H t o C H H 5 1 V 0 1 5 1 V 0 1 5 1 V 0 2 5 1 V 0 2 5 1 V 3 9 P 7 A 3 V C 3. A = V C 1.. B = V C n. A = V C h. B : V C . 1.. . a = H 1 o 8 3 1H 1 O : 3 1H 1 9 = 3 1H 1 O c. 3 1H 1 O : 3 : : H 1 2 1 VV 1 1 202 1 VV 1 702 1 VV 202 1 VV 1 702 1 VV 0110‘ 1 2 CC 6NN:0§0: CC 7NN010= CC BNNOQO: 9NN010: 0NN00910: C C2055: =32LLNT 0 0.5.2055: :32LLT 0 052055.. :32LLT 0 052055: :32LLT o .2055: :33LLT11315Q07 C:C: ..SF..C::C :=C=C= ..$F=C:= .. =C:C=:SF=C=: :2C:C=:C.F:C: : .. :C:C::$F:C:: :CC:C:C: BCBLNV VNVKRCOV'LBCBLNVVNV KROV1BC8LNVVNVKR OVCCBLNVVNVKHOV'LBCBLNVVNVKR ov a VIBCBL 111CC1FLN15CV1111LCC1F LNCCV111LCC1FLN CCV1111LCC1F LN 5CV111LCC1FLN GCIF V1111 8 9 O 1 2 2 2 3 3 3 5 5 5 5 5 620 625 630 635 6CD 6Q5 650 655 660 665 670 675 600 8 69‘: 9O SHYS VI-(LN-I) )=20E-3 o L 1 1 u N L C O 1 v 0 vv C11 CV = =C:C:=SF=C= .. 695 700 705 2NN°00 .- 5: :3LL7 01:2m55: .. 33LLT 6 552055: :33LLTEABR : : : : : 3T1... TY. I = =c:c- :SF 3 C = ..NTT.IBX‘DCH :N :1” 591 2 3 54‘ NVVNKROV18CBLNVVNVKROVIBCBLNVVNVKROHUHHVTTTTPOF LCC1LNGCV1111LCCIFLNGCV1111LCC1FLNC1111CAIll: C L.» . N 3 5 710 715 5HVL = 592656 CV CVOVlfiCLN-l) C11 7N13CCCC 602 12c 125 730 135 140 145 150 5 a 6 606 607 612 61! 755 760 765 770 9] 11 155 5.. 00° 6 CO. I .0. 1 .11 3 111 0 122 o 20. 1 .00 7 011 C :3 D 111 C C 125 O 5 111 1 1 CCC 2 I ODD O 1 0.. 00 Q 222 20 5 CCC ’1 6 ’1’ P1 2 111 P7 9 555 0 IC 5 11 0 1D 1 CCC 2 9 26 C DOD 0 1 .9 1 CCC Q :0 FCC 11 6 CCC O 1 1: ’1’ JJ 3 . u c 1 19 5° 081 JJ O 111 C 51 1o 1C5 00 1 CCC 5 1C CH P.0C 1011 01 CCC D 1 CD 51 1CD 12CC 2C I” cl 0 D. 11 10.25 CLL 1 111 I 0 ’1 C5 F5111101 1H 5.66 O a P1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 001 1 1 =17 o1C:HH :P 111AAACHHC 000000 PC 0 0 0 0 0 0 0 5. 91V: n. «J 0 SVOV.‘ V0V3V0°5V0 1.....16C72$1HPE1CCCCCTTTTTTOEQcoooof...» 3V3V3V3V3V3V3V3V3C3 3 3 3 333C3C3CC3C3V3UF1DCCC ::U =UDDDAAAAAAEU000000U2B C C C C C C C C 3 V V V V 8 2 = = :00 8 C NL:=:DTC11N11N:=8:82....:1":=====N== 0:0:0:0:0:0:0:0:010C0C0C0C0101010101VV010:01(111:N0111711111111111H1111111111 11?. 1111111111171191-Azl 31:1..12131 .15.! .CCT011119501113CCT3CT1801235551T123555T 12 2 3 5 5 6 7 8 D 1 1 1 1 1CC 1 9 "£11.15: 01" 1~12CCCCCCCNCCCCCCNCC OCOCOCOCOCOCOCOCOCOTOTOTOTOCOCOCOCOCFFOCOCOORCCCCJOHHOOHDCCCTTTTTTFOTTTTTTOEE CDCDCDCDCDCDCDCDCDCACA5A5AGDCDCDCDCSTSJCBCBCCFDDDSJDRPCDPCDDDAAAAAAolCAAAAAACBB 56789012345670.7012 341 001 .1 2 111112222222222333 330 577 2 n. 656665656555655555 553 333 Q Q 175 no 135 790 195 800 805 810 015 820 025 830 035 84“ 845 92 0‘161040PP0C0(4)002DOALOGCDCQDID¢3))I(D¢17)OD(1300(2)) DCITDODCI)0DC2D/(D(10)¢0(161.0(151) 1 1 , 5 1‘ 1 1D 1 1’ 1 51 3 ‘6 l. 11 0‘ D 11 ID I FF. 1‘ 1 1 1L1. 65 Q 1 2“ ‘0 1 3 CI! DL 0 1 (HM (I 1‘ f. CPD. 1115. 5 1B H151 78000 0 X! INS! 1121.6 L '1 11°F. ‘(oII‘ I 01111 03 ZSCR 0000/ 0 0111133331 0‘ C000 1/IIO110111119999111102‘71(11R 1PPP6926Q‘QQQCC((CC(((1XA1$FFF (PPP¢1(((((CCEEEEEEEEEEL‘lHLLR [too"E/EEEEE'YYYYYVYVEPX(O(‘( 82221DV1'v-VY' : : = : = = = : :9HLSCOOX .. = : : = : : = .. : : : :11111111) oCP .ZHHL 11111111111110123456789:HZ:RRP 00(16)0(D(Q)002)0ALOGCDC4)ID¢3))) )OYEC13)‘D(181/(2o00PP0(D(9)003)) (Q1002101L06¢DIQ)ID(31)]0‘11I 1 1 1 1 2 3 1 1 1 1 1 z 1 a! 1 4‘ o o 1 1 1 1 1 O 3 S 3 s 10 3° 1 1 1 1 1 1 1 o 1 O 5 O 5 O 1 2 Q 5 6 ‘3 l. 1 I. 1 61 61 2101‘ 1‘ cl 0| 1 [1151315131 9. 97 “(‘E E f— E f. 21‘7X1xYX1X—Jl‘ 21 I 0. 93 a B 8 B 1 ODD/v‘O’O‘g—L 5.1 1.10! 9 O O 0 1 1o 01 0.... .1 or. obv- 6: 00601 11 1 1 1 1 91°20'030'01‘ 1'- 1‘1‘1 02 ‘ 51 1 61 1 (2‘1‘1‘1‘l‘l‘2x 3‘ 1‘511111 1 1‘1 ‘1 (71811986 2. ISMSHSMSHECUEHUQCQCIc o o‘c‘TloCX QGGGXTGT GXT: : .0C5C9C5C9Co:~oCN(=C:LTTTGTG=LTGLT::=L::::L=11 : o: o: o: o: 0:V11V:IE1E1P : : : = = :19... :P:111P1111P1 1 3.56123.56789111111111/1CC1: :HZ13111311131KCTK1TY1Y2H1111113H11H‘1912H3Q56H72 D ((((((((((.I(((((((((((F1= =2IRC:3 . 0 c 5 c 5 . 7 . 8(1NC9N - 1 - 1C1.c..5781C23C9111C1111C1 1 EEEEEEEEEEEEEEEEEEEEEZ: (le\—PF : To! 2 It 2‘ 2 l. .- oI : (F‘OF‘O =1: ‘ .- ‘(l‘l‘o‘lt‘ 2“ = ‘1‘“: (((l‘ = (0‘ BBBB'YYV'VY'VYYYVVY'V'XBClBRRZHBlaxaxaxalgac'lacxaxBTGGGEGGVGGT6665756657500 850 855 860 865 070 875 000 005 890 095 900 905 910 915 92“ 93 1)‘Y(90LF)¢B(12))OB( )1) 0|: ‘1 1), ‘a .1 (F, \I F BLF F L 0 0L L O ,9 O 9 K 518 K ( «1“,... 1| '- OYY" '- O 2 O IFB O in ‘,)LO ’ )‘a J ‘87. '2 J F! O 9 O 07‘ O L I 2 .122" 2 3 I. a‘n‘vu’ I. 1‘ Q 1“ O) I (I O O 9 9’2 9 c. ’ 1”61 ) O, J ‘87 9‘ J )F 9 Q 9 028 0 IL .1 \a 1 011‘ O 1. O O ‘1 ‘1 ‘ "Q" n‘ 21 6 a I 8“ OF I ‘1‘ ‘ l‘ O 1 9 0.1L 0 IV 8 ‘1 a I. .1 9’" O \a I. o a I ‘0 I ‘1 J Q87 99 J ’F .1) .1 in, .1 6 ‘1’ a s! O ' O 1‘ 0 0L FF 0 18.. 2 C 62 ( 8Q 7 6 8 Q AQQn‘Y Q o 9 LL 0 Cl. 0 8 ‘0 a 1‘0 5 o 0 Q 9““ a! 01. O O 1 CE 0 I B. I HQ 2 Q Q A )ll 90 A 0‘ I! O '1'. ’ ‘1 I, \c ’9 9 0 0 I. 00.)) 9 0V (( o I. 5 5 ’7 7 )3 6 5 7 ‘1 1”6, ‘1 on VC 0 ‘1’) l. a. 5.! C 73 5 0 0 J 08? ’1 J 1) c o 1 "l, a a (a a ‘Q 2 Q Q 0 3 I ‘1. 9 (or ‘1) ‘ II... I. l. 8‘ Q 8) ‘1 \l ‘l ‘I 3 (33“ 3 val. I... ‘ I.“ \c 9’. 90,9 00 o o o 9 1| ““8 1| L. O. L 565 oggoQo,3HHO’80HHo o 00 0 80° 0 Dunno 1‘ I I‘ll 0.9 I P1. Q1 W ‘(Q canons-U O 0‘ o o 0‘ 05... occ 071‘ OCC .0 o O o O o o o o a o o. 89 ( U(()zs n! 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ILL ’00 266 0,), 8’0 0 0001 a 1UU O O .1. ’IUU 111‘ 1‘ c o 0 0 '8 1B), 800 ‘OFF 0 o 02,82LL 111,01 0 0 101,111,196 XXX)12,‘( LLL2111VC .PP92tbts1no HHH‘OB‘UTQ. CCCBZOBUT 0 :::((1(: :1 ,U’0’10,’-o FFF0010FF1 1005 1010 )) FJ 1 L 9 1 93 0 .01 s! 1‘ F C. L 0 ‘1 0 )F 1 JL 1‘ 0 0 N 2J V.) 0‘ (1 IV 1].? .- ’Q’L ,, ’9‘... 0 FF 115.! LL 19"( ’1. O O Eliot-5’10 JJ Y)’LF,F1 01 [FF QRFL 0 VC ’LLICL 009 O . F ' z‘ .1609 1) LBQNSI‘l 0 JJ 11Y81L19 0 0 8.1““LF09 Q1 11YY:CFR 09 (1 IY..:’:RC0: ll 1:1’F’Cl‘ o) ATANZCIIQXR10180IPP oEOoO)Q09oQ10 ): 9 IXXED 211901 90 810119Q12 1 0 £6 5‘..'~., 20) '2 .50001Q130Q1Q H R » 1: ) 11:2015310x.:14.1al.1x)» 0: C‘ .. =E=)FFLF15°F5FE’E 0’15 0c: EOEP’E 01 0E): 0175:1016 ‘l’UX—P LL OLL‘ELZLUSUZJI‘ZXBUII—LULGUZ‘ O U7U2‘UU .. NL‘ FFN Lviolflov 6 U o 1 5 U 5 1 2 2 o o o 1. 1 1 REILCUU) N ) ) 10XR10180IPP RT OLSFP : XIUP1D‘C’ 1)....NCl‘N x11’L1—DL L ONN CL! 9 ’ , I 0 91 1‘ 07 .L1( U :L 1N1m7|oT.HEANoVooN .‘N11IVMTINI$1NN1fLAu$JPn79LAl¥Af1lrkllntaflll LLL o 0‘. 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